How would I merge these 3 pipes with this corner?
Edit: Probably much easier since 2.8 with the new Mirror &quot;Bisect&quot; option
Former answer:
I've found an answer for a basic 3 cylinders shape, but it looks like it's not exactly what you're looking for. That said, I'm not sure your drawing means anything in 3D. Anyway, perhaps my solution will help?

Create a cylinder with Cap Fill Type &gt; Ngon.
In Edit mode, rotate it and move its cap to the origin with the Snap to Grid option (or put the 3D Cursor on the center of the scene, choose the 3D Cursor as Pivot, and S Y 0).
In Object mode duplicate the cylinder once and presse Enter.
Rotate the duplication 120° on the Z axis.
Duplicate and rotate again. Now you have a 3 branches star.
Join the 3 cylinders with 2 Booleans modifiers / Union mode.
Apply the Booleans.
Select all and W &gt; Remove Doubles.
Delete the inner edges.
Delete the useless vertices.
If you want to extend the cylinders length, you can select all the caps and use alt E &gt; Individual Faces, or select each cap and move it on its normal axis.